#506467

order	512
nilpotency class	4
isoclinism	isoclinic class of #500591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28280
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,15,18,29)(4,19)(5,9)(6,22)(7,12)(8,14)(10,17,26,32)(11,20)(13,21)(16,28,24,30)(23,25,31,27);(2,9,14,12)(3,16,18,24)(10,26)(11,13,22,19)(15,29)(17,31,32,23)(25,27)(28,30);(1,3)(2,10)(4,17)(5,16)(6,23)(7,24)(8,18)(9,25)(11,15)(12,27)(13,28)(14,26)(19,30)(20,31)(21,32)(22,29);(1,4,5,20,8,21,7,6)(2,11,12,19,14,22,9,13)(3,17,16,31,18,32,24,23)(10,15,27,30,26,29,25,28)

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413663);